# What does the space induced by this unusual metric(?) on R/Z look like?

The motivation for this question comes from music theory. Dmitri Tymoczko models "good" voice leading as minimizing distance between pitches in successive chords. While this theory works well for upper voices, it does not work so well for the bass, which tends to move by 4ths and 5ths quite frequently. (Tymoczko explicitly excludes the bass from his model.)

Taking log (all logs are base 2 in this question) of frequency and taking pitches which differ by an octave as equivalent, we get $\mathbb{R}/\mathbb{Z}$. For the upper voices, we want the standard metric on this.

For the bass, we want moving by a fifth or a fourth - meaning by $\pm \log (3/2)$ to be small. So we want $d(x,x\pm\log(3/2))=k_1$, where $k_1$ is probably somewhere around $0.05$. To make this a metric space, let's declare that $d(x,y)$ should be the minimum of $|x-y|$, $k_1+||x-y|-\log(3/2)|$, and $k_1+||x-y|-\log(4/3)|$.

We probably want moving by a major third - meaning by $\pm\log(5/4)$ to also be small, but not as small. I suspect $2k_1$ would make the most mathematical sense, but any constant of roughly that magnitude is fine. Ditto for minor thirds - this would be movement by $\pm\log(6/5)$, with a slightly larger constant.

If we do this, we might as well make all movements by $\pm\log(p/q)$ small if $q$ is small.

CLARIFICATION: I also want the standard metric to be one of the options for getting from $x$ to $y$. So the distance between $0$ and $\sqrt{1/500}$ should be $\sqrt{1/500}$, while the distance between $0$ and $7/12$ should be $k_1+|\log(3/2)-7/12|$. (Musically, $|\log(3/2)-7/12|$ is how far off an even-tempered 5th is from Pythagorean tuning.)

Question 1: Can one actually define something along these lines that satisfies the triangle inequality? (I don't think I actually have; I probably need to take the minimum (or infimum) of some infinite sequence, but am not entirely sure that works.)

Question 2: Assuming the answer to (1) is yes, what does this metric space look like? Can someone help me with a picture that seems less exotic, perhaps comparing it to the Hawaiian Earring or something of that sort? In particular, what might the fundamental group look like?

My background: I'm a combinatorialist and algebraic geometer who happens to be the one least unqualified here to be supervising an undergraduate independent study on mathematics in music theory. I did the standard first year graduate courses in point set topology and algebraic topology, but that was almost a dozen years ago.

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You haven't defined what happens with irrational multiples, but I suspect you want them to be more distant than low-height rationals. This seems to force your space to be an uncountable discrete set. –  S. Carnahan May 11 '11 at 6:31

Let me give a try for tempered scales. (In a tempered scale, the ratio of frequencies for adjacent halftones is always $2^{1/(12)}$.)

In this case, we want a distance on $\mathbb R/\mathbb Z$ (with corresponding frequencies given by $2^x/2^{\mathbb N}$, invariant under rotations such that $\frac{5}{12}$ is close to $0$ (in $\mathbb R/\mathbb N$). Say we want it to be closer to $0$ than any other element among $\frac{1}{12},\frac{2}{12},\frac{3}{12},\frac{4}{12},\frac{6}{12}$. One possible recipee: Choose strictly positive rationals $\alpha_1,\dots,\alpha_6\in (0,1/12]$ encoding pleasantness of intervals (say $\alpha_5<\alpha_4<\alpha_3<\alpha_2<\alpha_1<\alpha_6$) and set $d(y,x)=d(x,y)=\min(y-x,\alpha_i+\vert y-x-i/12\vert)$ if $y>x$ with $y-x\leq 1/2$.

The graph of the distance function is a symmetric function modelling the skyline of a symmetric island consisting entirely of montains with slope $1$. Pleasant intervals correspond to passes.

Remark: The requirement $\alpha_i\leq 1/12$ is slightly to strong: we need only $\alpha_i\leq \alpha_j+\vert i-j \vert/{12}$.

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A perhaps pleasant choice is given by $\alpha_5=2\epsilon$, $\alpha_4=3\epsilon$, $\alpha_2=4\epsilon$, $\alpha_3=5\epsilon$, $\alpha_1=6\epsilon$, $\alpha_6=7\epsilon$ where $\epsilon$ is small and positive. Justification: One wants a large (relative) distance if one is out of pitch (this is achieved by $\epsilon$ small). A more or less "rational" choice of distance is then to consider a distance proportional to $a+b$ if an interval consists of two notes with frequencies (roughly) proportional to $a/b2^{\mathbb Z}$. –  Roland Bacher May 11 '11 at 12:18
These choices don't satisfy the triangle inequality: $d(0,1/12)>d(0,5/12)+d(1/12,5/12)$. –  Alexander Woo May 11 '11 at 15:47
Right. Add $\epsilon$ everywhere. (I guess one can find a nice way related to proportions but this requires probably some trial and error. My first attempt was too naive.) –  Roland Bacher May 11 '11 at 17:52

I doubt that you can do this in a way which addresses the intended application. For instance, you say you want bass movement by a fifth to be "small" because it is frequent in traditional music. You also want stepwise movement by a full or half step to be "small." But you can combine a fifth and a half step to make a tritone, and the triangle inequality would then force that to be "small." But this is not something you want. It seems to me that the triangle inequality is not an adequate assumption to impose on what you are trying to model.

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I think it is right for tritones to be not too big; bass movement by tritone does happen with some frequency, usually harmonically from the tonic to the dominant of the dominant in first inversion. –  Alexander Woo May 11 '11 at 19:06
Another obvious flaw is that, if you take notes that differ by an octave as equivalent, then stepwise motion is equivalent to leaps by sevenths. It is far from so in real music. –  Michael Renardy May 11 '11 at 21:57